In this article, we study the existence of solutions to first order non-linear initial value problems within the field of “dynamic equations on time scales”

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF SOLUTIONS AND CONVERGENCE RESULTS FOR DYNAMIC INITIAL VALUE PROBLEMS USING LOWER

AND UPPER SOLUTIONS

ATIYA H. ZAIDI

Abstract. In this article, we study the existence of solutions to first order non-linear initial value problems within the field of “dynamic equations on time scales”. We employ the method of upper and lower solutions and Schauder’s fixed point theorem. We also provide sufficient conditions under which the upper and lower solutions converge uniformly to a solution. Some examples are given to illustrate the new results.

1. Introduction

Dynamic equations on time scales had been introduced in 1988 as generalised forms of mathematical modelling which can incorporate the structure of differential or difference equations or both at the same time, see [12, 13, 1].

This article considers the dynamic initial value problem

x^∇=f(t, x), for allt∈[0, a]_κ,_T; (1.1)

x(0) = 0. (1.2)

Here x^∇ is the “nabla” derivative ofxintroduced in [8, p.77]. Here f : [0, a]κ,T× [l, u]⊂R²→Ris a left-Hilger-continuous and possibly non-linear function andl, u are continuous on [0, a]_T= [0, a]∩Tfor an arbitrary time scaleT. The subscriptκ refers to [0, a]_Tless any right scattered minimum points in it [9, p.331]. The term

“left-Hilger-continuous” is used in accordance with the term “left-dense-continuous”

(or ld-continuous) [9, Definition 8.43] and will be defined in the next section.

Our results show that (1.1), (1.2) has at least one solution which is bounded above by some function,u, and is bounded below by another function,l, whereu, l are upper and lower solutions to (1.1), (1.2). The results follow some notions of La Salle [16] extended to the time scale setting. In this way, our results exhibit a broader span of modelling a system described as a first order initial value problem, no matter if the system has a discrete or a continuous domain or a hybrid of both.

We apply our ideas to establish non-negative solutions to (1.1), (1.2).

2000Mathematics Subject Classification. 34N05, 26E70.

Key words and phrases. Existence of solutions; non-linear nabla equations;

lower and upper solutions; zero approximations.

c

2009 Texas State University - San Marcos.

Submitted November 24, 2009. Published December 17, 2009.

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In addition, we establish sufficient conditions under which: solutions to (1.1), (1.2) established within [l, u] are unique; andlanduapproximate solutions to (1.1), (1.2). Then we establish an error estimate on thei-th approximation.

The motivation for using upper and lower solutions in our results are due to the wide use of this method to establish existence results for a variety of first and second order initial and boundary value problems, see [3, 4, 5, 6, 7, 8, 9, 10, 14, 21].

In this work, we use this method to determine: existence of solutions to (1.1), (1.2);

and establishing successive approximations converging to a solution of the above IVP.

It had been shown in [6] that existence results involving lower and upper solutions in the time scale setting can be proved with less restrictions using nabla derivatives than using delta derivatives. We use nabla derivatives in this work to allow the solution to assume maximal values at the right end point of a given interval of existence, [l, u], using the maximum principle. In this way, our results are different from the existence and uniqueness results for the first order IVPs involving delta derivatives proved in [20] using fixed point theorems and in [19] using the method of successive approximations. Our results are also different in context and methodology from the existence and uniqueness results using lower and upper solutions for the first order delta IVPs proved in [15, Theorem 4.1.2].

This paper is organised in the following manner. In Section 2, a brief introduction to the time scale calculus concerning nabla derivatives is presented. For more details, see [8, pp.77–81] and [9, Chapter 1, Chapter 8].

In Section 3, we define lower and upper solutions to the dynamic IVP (1.1), (1.2) and establish existence and uniqueness of solutions to (1.1), (1.2) within lower and upper solutions to the IVP.

In Section 4, we show thatl(t), u(t) are zero approximations to solutions of (1.1), (1.2) established in Section 3, for allt∈[0, a]_T. We also prove that an upper bound exists on the error of thei-th approximation on [0, a]_Twhich approaches to zero for a unique solution.

2. Preliminaries

A time scale, denoted byT, is a non-empty closed subset ofR. Thus,N,Z, [0,1], [−1,0]∪[2,5] and the Cantor set are examples of time scales. A dynamic equation on time scales models a phenomenon that may be continuous at one time and discrete at another. Hence, those dynamic equations that demonstrate a completely continuous phenomenon (or alternatively a completely discrete phenomenon) are equivalent to a differential equation (alternatively a difference equation).

For any pointt∈T, the left and right movements are measured in terms of left and right “jump operators”, named “ρ(t)” and “σ(t)” respectively. These operators are defined as

ρ(t) := sup{s∈T:s < t}, for allt∈T; σ(t) := inf{s∈T:s > t}, for allt∈T.

We can see from the above inequalities thatρ(t) andσ(t) would coincide witht in continuous intervals ofT. Within discrete intervals ofT, the functions

ν(t) :=t−ρ(t), for allt∈Tκ; µ(t) :=σ(t)−t, for allt∈T^κ,

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where T^κ refers to T less any left-scattered maximum points in it [12, p. 27], describe a measure of the step size between two consecutive points. The point tis considered “left-scattered” whenν(t)>0 and “left-dense” whenν(t) = 0. Similar relationships hold between the appearance of t to the right and µ(t). Our results in this work concern the behaviour of points to the left of t. Hence, further ideas regard left-dense or left-scattered points only.

All continuous functions on a time scale are ld-continuous [9, Theorem 8.43].

The term “left-Hilger-continuous” introduced in the previous section forf in (1.1) is used in equivalence with the term “ld-continuous” for a function of two or more variables, the first of which should be from an arbitrary time scale. This is a more generalised definition and we have introduced this particular term for functions of several variables, to avoid confusion with ld-continuous functions of one variable.

Definition 2.1(Left-Hilger-continuous functions). A mappingf : [a, b]_κ,_T×R→R is called left-Hilger-continuous at a point (t, x) if: f is continuous at each (t, x) wheret is left-dense; and the limits

lim

(s,y)→(t⁺,x)

f(s, y) and lim

y→xf(t, y) both exist and are finite at each (t, x) where t is right-dense.

The following definitions and theorem [9, Section 8.4] describe nabla differentiable functions and their properties for a generalised time scale and will be funda- mental to our results in this work.

Definition 2.2 (The nabla derivative). Let x : T → R and t ∈ Tκ. Define x^∇(t) to be the number (if it exists) with the property that given >0 there is a neighbourhoodN oftwith

|[x(ρ(t))−x(s)]−x^∇(t)[ρ(t)−s]| ≤|ρ(t)−s|, for alls∈N.

We call x^∇(t) the nabla derivative ofx(t) for all t ∈ T^κ and say that xis nabla differentiable onT^κ.

Theorem 2.3. LetTbe an arbitrary time scale and consider a functionh:Tκ→R. Then the following hold for all t∈Tκ:

(1) ifhis nabla differentiable att, thenhis continuous att;

(2) if his continuous at t andt is left-scattered, then his nabla differentiable att and

h^∇(t) := h(t)−h(ρ(t)) ν(t) ;

(3) ift is left-dense, then his nabla differentiable attsuch that h^∇(t) :=lims→t

h(t)−h(s) t−s ,

provided the limit on the right hand side exists and is finite;

(4) ifhis nabla differentiable att then

h^ρ(t) :=h(t)−ν(t)h^∇(t), whereh^ρ=h◦ρ.

Hence, ifT=Rthenx^∇=x⁰, while ifT=Zthenx^∇=∇x(t) =x(t)−x(t−1).

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Definition 2.4 (The nabla integral). Leth:T→R. A function H :T→Rwill be a nabla anti-derivative ofhifH^∇(t) =h(t) holds for allt∈Tκ. Lett0∈Twith t0< tthen the Cauchy nabla integral of his defined as

Z t t₀

h(s)∇s:=H(t)−H(t0), for allt∈T.

3. Existence results

In this section, we define lower and upper solutions to (1.1), (1.2). We also prove that (1.1), (1.2) has a solution on [0, a]_T that lies within the interval [l, u], where l(t), u(t) act respectively as lower and upper solutions to (1.1), (1.2) for all t∈[0, a]_T, using Schauder’s fixed point theorem.

Definition 3.1. Letl, u be nabla differentiable functions on [0, a]κ,T. We calll a lower solution to (1.1), (1.2) on [0, a]_T if

l^∇(t)≤f(t, l(t)), for allt∈[0, a]_κ,_T; (3.1)

l(0) = 0. (3.2)

Similarly, we calluan upper solution to (1.1), (1.2) on [0, a]_T if

u^∇(t)≥f(t, u(t)), for allt∈[0, a]_κ,_T; (3.3)

u(0) = 0. (3.4)

Definition 3.2. A solution of (1.1), (1.2) is a nabla differentiable function x : Tκ→Rthat satisfies (1.1) and (1.2) and the point (t, x(t))∈[0, a]_T×[l, u], where l, uare continuous on [0, a]_T.

All ld-continuous functions are nabla integrable [9, Theorem 8.45]. The following lemma establishes equivalence of (1.1), (1.2) as nabla integral equations. The result is nabla equivalent of ideas in [20, Lemma 2.1] for the “delta” case. Therefore, the proof is omitted.

Lemma 3.3. Consider the dynamic IVP (1.1),(1.2). Let f : [0, a]κ,T×[l, u]→R be a left-Hilger-continuous function. Then a functionx∈C([0, a]_T;R)solves (1.1), (1.2)if and only if it satisfies the nabla integral equation

x(t) = Z t

0

f(s, x(s))∇s, for allt∈[0, a]_T. (3.5) The following definition and the next two theorems are the keys to our proof for the existence of solutions to (1.1), (1.2).

Definition 3.4. [22, p.54] Let U, V be Banach spaces andF : A⊆U →V. We sayF is compact onAif:

• F is continuous onA;

• for every bounded setB ofA,F(B) is relatively compact inV.

The next theorem from [18, Theorem 1.3] is stated in the context ofT⊆R. The proof is, therefore, omitted.

Theorem 3.5 (Arzela-Ascoli theorem on T). Let D ⊆C([a, b]_T;R). Then D is relatively compact if and only if it is bounded and equicontinuous.

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Theorem 3.6 (Schauder’s fixed point theorem [17, p.67] [22, p.57]). Let X be a normed linear space and D be a closed, bounded and convex subset of X. If F :D→D is a compact map then F has at least one fixed point.

Define an infinite strip

S_κ,∞:={(t, p) :t∈[0, a]κ,Tand − ∞< p <∞}.

Letg:Sκ,∞→Rbe a left-Hilger-continuous function. Our next theorem concerns the existence of solutions to the initial value problem

x^∇=g(t, x), for allt∈[0, a]_κ,_T; (3.6)

x(0) = 0 (3.7)

inSκ,∞. We prove this result by using Schauder’s fixed point theorem.

Theorem 3.7. Consider the initial value problem (3.6), (3.7) with g left-Hilger- continuous on S_κ,∞. If g is uniformly bounded on S_κ,∞ then (3.6), (3.7) has at least one solution,x, such that the point(t, x(t))lies in the infinite strip

S_∞:={(t, p) :t∈[0, a]_T and − ∞< p <∞}.

Proof. From Lemma 3.3, a solution of (3.6), (3.7) is given by x(t) :=

Z t 0

g(s, x(s))∇s, for allt∈[0, a]_T. (3.8) Sinceg is uniformly bounded onS_κ,∞, there existsM >0 such that

|g(t, p)| ≤M, for all (t, p)∈S_κ,∞. (3.9) Define K := M a and consider the Banach space (C([0, a]_T;R),| · |0) [20, Lemma 3.3]. LetD⊂C([0, a]_T;R) defined by

D:={x∈C([0, a]_T;R); |x|0≤K}.

ThenD is closed, bounded and convex. We show that a compact mapF :D→D exists and Schauder’s theorem applies.

Define

[F x](t) :=

Z t 0

g(s, x(s))∇s, for allt∈[0, a]_T. (3.10) SeeF is well defined onC([0, a]_T;R) asg is left-Hilger continuous onS_κ,∞.

We show thatF :D→Dis a compact map. For this, we show that the following properties hold forF:

(i) F is continuous onD;

(ii) for every bounded subsetBofD,F(B) is relatively compact inC([0, a]_T;R), and verify Definition 3.4.

To show thatF is continuous onD, we define BK(0) :={p∈R:|p| ≤K}.

SeeBK(0) is closed and bounded and hence compact inR. Therefore,gis bounded and uniformly left-Hilger-continuous on [0, a]_T×BK(0). Thus, for every 1 > 0 there exists aδ1=δ1(1) such that for (t, x1),(t, x2)∈[a, b]κ,T×BK(0), we have

|g(t, x1)−g(t, x₂)|< ₁ whenever|x1−x₂|< δ₁. (3.11)

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Letxi be a convergent sequence in D withxi →xfor alli. Then for everyδ1>0 there exists N >0 such that

|xi−x|< δ1, for alli≥N.

We show that the sequenceF_i:=F x_i is uniformly convergent inR. Let₀:=₁a.

We see that

|F xi−F x|0= sup

t∈[0,a]T

|F xi(t)−F x(t)|

≤ sup

t∈[0,a]T

Z t 0

(g(s, xi(s))−g(s, x(s)))∇s

≤ sup

t∈[0,a]_T

Z t 0

|g(s, xi(s))−g(s, x(s))|∇s

< 1a whenever|xi−x|< δ1

=0,

for all i ≥ N. Thus Fi are uniformly convergent on D and hence are uniformly continuous onD. We show thatF :D→D: See for allx∈D, we have

|F x|0:= sup

t∈[0,a]T

|F x(t)|

≤ sup

t∈[0,a]_T

Z t 0

|g(s, x(s))|∇s

≤M a=K.

(3.12)

Thus,F is inD.

Next, we show that for every bounded subsetBofD,F(B) is relatively compact onC[0, a]_Tusing the Arzela-Ascoli theorem. LetBbe an arbitrary bounded subset of D. Assume x∈ B. Then we see from (3.12) that we have |F x|₀ ≤K for all t∈[0, a]_T. ThusF is uniformly bounded onB.

We also see that for any given >0 we can defineδ:=/Mand fort₁, t₂∈[0, a]_T, we obtain

|[F x](t₁)−[F x](t₂)|=

Z t2

t1

g(s, x(s))∇s

≤

Z t2

t₁

|g(s, x(s))|∇s

≤M |t1−t2|<

whenever|t1−t2|< δ. Hence,F is equicontinuous. By the Arzela-Ascoli theorem, F(B) is relatively compact inC([a, b]_T;R).

From (i) and (ii) above, we see thatF :D→D is a compact map. We also see thatF satisfies the conditions of Schauder’s theorem and, so, has at least one fixed point inD given by (3.8). Hence, (3.6), (3.7) has at least one solution,x, such the

point (t, x(t))∈S∞.

The above result ensures existence of a solution to (3.6), (3.7) when the function gis bounded in an infinite domainSκ,∞and considers this as a sufficient condition for the existence of a solution to the above IVP inS_κ,∞. However, the result does not ensure the existence if the domain is restricted.

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In the next result, we strengthen the condition in the above theorem by restrict- ing the solution to (3.6), (3.7) within a lower and an upper solution to (1.1), (1.2).

Hence we prove the existence of a solution to (1.1), (1.2) within the region S :={(t, p) :t∈[0, a]_Tandl(t)≤p≤u(t)},

wherel, uare, respectively, lower and upper solutions to (1.1), (1.2). To prove this, we define a modified functiongin terms off in (1.1) and prove thatgis uniformly bounded onS_κ,∞ and use Theorem 3.7. We also prove that the solution,x, to the IVP (3.6), (3.7) satisfiesl(t)≤x(t)≤u(t) for allt∈[0, a]_T, so thatxmust also be a solution to the original unmodified problem (1.1), (1.2).

Define

S_κ:={(t, p) :t∈[0, a]_κ,_T andl(t)≤p≤u(t)}.

Theorem 3.8. Let f : Sκ → R be a left-Hilger-continuous function. If l, u are, respectively, lower and upper solutions to (1.1),(1.2), then (1.1),(1.2)has at least one solution,x, such thatl(t)≤x(t)≤u(t)for all t∈[0, a]_T.

Proof. Consider the IVP (3.6), (3.7), whereg(t, p) is defined onS_κ,∞ such that for allt∈[0, a]κ,T,

g(t, p) :=











f(t, l(t)) + l(t)−p

1 + (l(t)−p)², ifp < l(t);

f(t, p), ifl(t)≤p≤u(t);

f(t, u(t))− p−u(t)

1 + (p−u(t))², ifp > u(t).

(3.13)

We first show thatgis left-Hilger-continuous and uniformly bounded onSκ,∞and Theorem 3.7 applies. Note that f is left-Hilger-continuous on the compact region S_κ and so it is bounded onS_κ. Thus, there existsM₁>0 such that|f(t, p)| ≤M₁ for all (t, p)∈S_κ. We also see that forl(t)> p∈R, we have

l(t)−p 1 + (l(t)−p)²

<1, for allt∈[0, a]_T, and so

f(t, l(t)) +

l(t)−p 1 + (l(t)−p)²

<1 +M1, for allt∈[0, a]κ,T. LetM := 1 +M1. Then from (3.13), we obtain

|g(t, p)| ≤M, for all (t, p)∈S_κ,∞. (3.14) Hence g is uniformly bounded onS_κ,∞. In addition, the left-Hilger-continuity of f on Sκ and the ld-continuity of l, u, p on [0, a]_T show that the right hand side of (3.13) is left-Hilger-continuous on S_κ,∞ and, so, we have g left-Hilger-continuous on Sκ,∞. By Theorem 3.7, the modified IVP (3.6), (3.7) has a solution, x, such that the graph (t, x(t))∈S∞ for allt∈[0, a]_T.

Next, we prove thatl(t)≤x(t)≤u(t) for allt∈[0, a]_T. We split the inequality l(t)≤x(t)≤u(t) into two parts and first show that

l(t)≤x(t), for allt∈[0, a]_T, (3.15) using the contradiction method.

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Letr(t) :=l(t)−x(t) for allt∈[0, a]_T. Assume there exists a pointt1 ∈[0, a]_T such thatl(t1)> x(t1). Seet16= 0 asx(0) = 0 =l(0) from (1.2) and (3.2). Without loss of generality, we may assume that

r(t1) = max

t∈[0,a]_Tr(t)>0. (3.16)

Thus,r(t) is non-decreasing att=t₁and, so,r^∇(t₁)≥0.

On the other hand, sincex(t1)< l(t1) we see that using (3.6), (3.13) and (3.1), we obtain

0≤r^∇(t1) =l^∇(t1)−x^∇(t1)

=l^∇(t₁))−g(t₁, x(t₁))

=l^∇(t1))−f(t1, l(t1))− l(t1)−x(t1) 1 + (l(t₁)−x(t₁))²

< l^∇(t₁))−f(t₁, l(t₁))≤0,

which is a contradiction. Hence l(t)≤x(t) for all t∈[0, a]_T. It is very similar to show thatu(t)≥x(t) for allt∈[0, a]_T as in the above case. We omit the details.

Thus, we havel(t)≤x(t)≤u(t) for allt∈[0, a]_T. Hence, from (3.13),x(t) is a solution to (1.1), (1.2) for allt∈[0, a]_Tand the point (t, x(t))∈S for allt∈[0, a]_T.

This completes the proof.

The following example illustrates the above theorem.

Example 3.9. Consider the Riccati initial value problem

x^∇(t) =f(t, x) :=x²−t, for allt∈[0,1]_κ,_T; (3.17)

x(0) = 0. (3.18)

We claim that there exists at least one solution, x, to the above IVP such that

−t≤x(t)≤t for allt∈[0,1]_T.

Proof. We see that the right hand side of (3.17) is a composition of a continuous functiontand a continuous functionx²and hence, is continuous on [0,1]_T×R. So ourf is left-Hilger-continuous on [0,1]κ,T×R. Let us define

l(t) :=−t, for allt∈[0,1]_T. Then we see thatl(0) = 0 and for allt∈[0,1]_T, we have

f(t, l(t)) =t²−t≥ −1 =l^∇(t).

Thus, ourlsatisfies (3.1), (3.2) and is a lower solution to (3.17), (3.18).

In a similar way, the functionu(t) :=tis an upper solution to (3.17), (3.18) for allt ∈[0,1]_T. By Theorem 3.8, there is at least one solution, x, to (3.17), (3.18)

such that−t≤x(t)≤tfor allt∈[0,1]_T.

Our next result gives a sufficient condition for uniqueness of solution to (1.1), (1.2). We show that the solution,x, of the above IVP established in Theorem 3.8 is the only solution satisfyingl(t)≤x(t)≤u(t) for allt∈[0, a]_T.

Theorem 3.10. Let f be left-Hilger-continuous on Sκ. Assume l, u are, respectively, lower and upper solutions of (1.1),(1.2). If there existsL >0 such thatf satisfies

|f(t, p)−f(t, q)| ≤L|p−q|, for all (t, p),(t, q)∈S_κ, (3.19)

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then the solutionxof (1.1),(1.2)brought forward under the conditions of Theorem 3.8 is the unique solution satisfyingl(t)≤x(t)≤u(t)for all t∈[0, a]_T.

Proof. Letx, ybe solutions of (1.1), (1.2) such that the points (t, x(t)),(t, y(t))∈ S_κ. Then, using (3.5), we obtain for allt∈[0, a]_T,

|x(t)−y(t)| ≤ Z t

0

|f(s, x(s))−f(s, y(s))|∇s

≤L Z t

0

|x(s)−y(s)|∇s,

(3.20)

where we employed (3.19) in the last step. Define

k(t) :=|x(t)−y(t)|, for allt∈[0, a]_T.

See, L >0 and so L∈ L⁺ [7, p.225]. Applying Gronwall’s inequality concerning nabla derivatives [7, Theorem 2.7] (taking f(t) = 0 and p(t) = L) to (3.20), we obtain

k(t)≤0, for allt∈[0, a]_T.

Butk(t) =|x(t)−y(t)|and so, is non-negative for allt∈[0, a]_T. Thus,x(t) =y(t)

for allt∈[0, a]_T.

The next corollary establishes existence of a unique, non-negative and bounded solution of the IVP (1.1), (1.2) on [0, a]_T.

Corollary 3.11. Let f : Sκ → R be a left-Hilger-continuous function satisfying (3.19). Let l, u be lower and upper solutions to (1.1),(1.2). Ifl(t) = 0 for all t∈ [0, a]_T, then the IVP (1.1),(1.2) has a unique, bounded and non-negative solution, x(t), for allt∈[0, a]_T.

The proof of the above corollary follows from Theorem 3.10, as 0≤x(t)≤u(t) for allt∈[0, a]_T. The following example illustrates this result.

Example 3.12. Consider the dynamic initial value problem

x^∇(t) =f(t, x) :=ρ(t) +x³, for allt∈[0,1]κ,T; (3.21)

x(0) = 0. (3.22)

We claim that the above IVP has a unique non-negative solution xsuch that 0≤ x(t)≤1 for all t∈[0,1]_T.

Proof. See f(t, p) = ρ(t) +p³ for all (t, p) ∈ [0,1]κ,T×R. Since ρ(t) and p³ are everywhere ld-continuous functions and so is their composition, ourf is left-Hilger- continuous on [0,1]κ,T×R. We define

l(t) := 0, and u(t) :=t², for allt∈[0,1]_T.

Then we see that l(t)≤u(t) for allt ∈[0, a]_T withl(0) = 0 =u(0). It is evident that l satisfies (3.1) and so, is a lower solution to (3.21), (3.22). We also see that, for allt∈[0,1]_T

f(t, u(t)) =ρ(t) +t⁶≤ρ(t) +t=u^∇(t).

Thus, ourusatisfies (3.3) and is an upper solution to (3.21), (3.22). By Theorem 3.8, there exists a solution,x, to (3.21), (3.22) such that 0≤x(t)≤t²≤1, for all t∈[0,1]_T. Moreover, for all t∈[0,1]_T, we have

∂f

∂p

=|3p²| ≤3t⁴≤3.

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Thus, f has bounded partial derivatives in [0,1]_T×[0,1] and satisfies (3.19) for L = 3 (see [2, Lemma 3.2.1], [11, p.248]). By Corollary 3.11, x is the unique solution to (3.21), (3.22) such that 0≤x(t)≤1 for allt∈[0,1]_T.

4. Convergence results

In this section, we establish conditions under which lower and upper solutions to (1.1), (1.2) approximate the existing solutions of (1.1), (1.2). We also establish error estimates on heith approximation.

Letf :Sκ→Rbe left-Hilger-continuous. DefineF :C([0, a]_T;R)→C([0, a]_T;R) by

[F p](t) = Z t

0

f(s, p(s))∇s, for allt∈[0, a]_T.

Then F is well-defined on C([0, a]_T;R). Under the conditions of Theorem 3.8, a fixed pointxofF will be a solution to (1.1), (1.2) such thatl(t)≤x(t)≤u(t) for allt∈[0, a]_T, wherel, uare, respectively, lower and upper solutions of (1.1), (1.2).

Consider an iterative scheme defined as [F⁰p](t) := [F p](t) =

Z t 0

f(s, p(s))∇s, for allt∈[0, a]_T; (4.1) Fⁱ :=F[Fⁱ⁻¹], for alli≥1. (4.2) It had been shown in [19, pp.78–79] that, in general, the continuity of a functionf alone is not sufficient for a sequence or subsequences of successive approximations to converge to a solution on a compact rectangle. In our next result, we show that the successive approximations defined in (4.1), (4.2) provide a sequence of functions that converge to a solution to (1.1), (1.2).

We assumef to be non-decreasing onSκand prove that ifxis a solution to (1.1), (1.2) such thatl(t)≤x(t)≤u(t) for allt∈[0, a]_T, thenl(t) andu(t) approximate x(t) for allt∈[0, a]_T. We also show that an upper bound on the error of the ith approximation will be [Fⁱu](t)−[Fⁱl](t) for all t ∈ [0, a]_T. The next definition describes zero approximation to the solution of (1.1), (1.2) (see [16, p.724] for the ODE case).

Definition 4.1. Letxbe a solution to (1.1), (1.2) andy:T→Rbe a ld-continuous function. We call y(t) a zero approximation to x(t) for all t ∈ [0, a]_T if, {Fⁱy}

converges uniformly toxon [0, a]_T.

Theorem 4.2. Let f : S_κ → R be left-Hilger-continuous and l, u are lower and upper solutions to (1.1), (1.2). If f is non-decreasing in the second argument on Sκ, that is, forp≤q, we have

f(t, p)≤f(t, q), for all(t, p),(t, q)∈S_κ; (4.3) thenl(t)andu(t)will be the zero approximations to a solutionxof (1.1),(1.2)for allt∈[0, a]_T.

Moreover, form, n≥0, the sequence Fⁱ given by (4.1),(4.2)satisfies

[F^ml](t)≤[F^m+1l](t)≤[Fⁿ⁺¹u](t)≤[Fⁿu](t), for allt∈[0, a]_T. (4.4) Proof. We show thatl(t), u(t) satisfy Definition 4.1 and (4.4) holds for allt∈[0, a]_T. We see from (4.1) that forp=u, we obtain for allt∈[0, a]_T

[F u](t) = Z t

0

f(s, u(s))∇s≤ Z t

0

u^∇(s))∇s=u(t). (4.5)

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Similarly, forp=l, we obtain

l(t)≤[F l](t) for allt∈[0, a]_T. (4.6) Sincef is non-decreasing in the second variable and is left-Hilger-continuous onSκ, it follows from (4.1), (4.6), and (4.3) that, for allt∈[0, a]_T, we have

(t) = [F l](t) = Z t

0

f(s, l(s))∇s

≤ Z t

0

f(s,[F l](s))∇s

= [F¹l](t).

(4.7)

Proceeding in this way, we obtain

[F l](t)≤[F¹l](t)≤[F²l](t)≤[F³l](t)≤. . . , for allt∈[0, a]_T. (4.8) Thus, the sequence{Fⁱl}is non-decreasing. In a similar way, using (4.3), (4.1) and (4.5), we obtain

[F u](t)≥[F¹u](t)≥[F²u](t)≥. . . , for allt∈[0, a]_T. (4.9) Now sincel(t)≤u(t) for allt∈[0, a]_κ, we can write using (4.8) and (4.9) that for allt∈[0, a]_κ

[Fⁿl](t)≤[Fⁿ⁺¹l](t)≤[Fⁿ⁺¹u](t)≤[Fⁿu](t). (4.10) We further see that

[F l](0) = 0 = [F u](0). (4.11)

We show that the sequence{Fⁱl}converges uniformly to the fixed pointx(x(0) = 0). Define

w(t) := [F u](t)−[F l](t), for allt∈[0, a]_T.

See,w(t)≥0 for allt∈[0, a]_T. Sincef is non-decreasing in the second variable on S_κ, it follows from (4.1) that

w^∇(t) =f(t, u(t))−f(t, l(t))≥0, for allt∈[0, a]_κ,_T. It is clear from (4.10) that forn > m≥0, we have

[F^ml](t)≤[Fⁿl](t)≤[Fⁿu](t), and forn < m, we have

[F^ml](t)≤[F^mu](t)≤[Fⁿu](t).

Hence for anym, n≥0, we have the inequality

[F^ml](t)≤[F^m+1l](t)≤[Fⁿ⁺¹u](t)≤[Fⁿu](t), for allt∈[0, a]_T.

The boundedness and equicontinuity of eachFⁱl can be established in the same way as in Theorem 3.7. Hence, asi→ ∞,Fⁱl converges uniformly on [0, a]_T to a fixed point x. Similarly, {Fⁱu} converges uniformly on [0, a]_T to a fixed pointx.

Thusl(t) andu(t) are zero approximations tox(t) withwⁱ(t) := [Fⁱu](t)−[Fⁱl](t) as an upper bound on the error of thei-th approximation for allt∈[0, a]_T. If the solution is unique, then wⁱ(t)→ 0 for all i≥1 for all t∈[0, a]_T. This completes

the proof.

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Example 4.3. Consider the dynamic IVP

x^∇(t) =f(t, x) :=x³−t, for allt∈[0,1]κ,T; (4.12)

x(0) = 0. (4.13)

We claim that l(t) =−t and u(t) = t are zero approximations to the solution of (4.12), (4.13) for allt∈[0,1]_T. Moreover, for all t∈[0,1]_T, the sequenceFⁱ given by

F⁰(t) := [F x](t) = Z t

0

(x³−s)∇s, Fⁱ :=F[Fⁱ⁻¹], for alli≥1.

satisfies (4.4) for anym, n≥0.

Proof. We see that f(t, p) = p³−t for all (t, p) ∈ [0,1]κ,T×R. Since t and p³ are everywhere ld-continuous functions and so is their composition, our f is left- Hilger-continuous on [0,1]κ,T×R. We further see that l(0) = 0 =u(0) and for all t∈[0,1]_T

f(t, l(t)) =−t(t²+ 1)≥ −1 =l^∇(t).

Thus, l satisfies (3.1) and so, is a lower solution to (4.12), (4.13). In a similar way, we have usatisfying (3.3) and so, is an upper solution to (4.12), (4.13). By Theorem 3.8, there exists a solution,x, to (4.12), (4.13) such that−t ≤x(t)≤t, for allt∈[0,1]_T.

Next, we see that forp≤q, we have

f(t, p) =p³−t≤q³−t=f(t, q), for all (t, p),(t, q)∈[0,1]_κ,_T×[−t, t].

Thus,f is non-decreasing with respect to the second argument on [0,1]_T,κ×[−t, t]

and so, by Theorem 4.2, the functions −t and t are zero approximations to the solutionxof (4.12), (4.13). We further see that forx=l, we have for allt∈[0,1]_T,

[F l](t) = Z t

0

−(s³+s)∇s≥ −t=l(t).

This leads to (4.7) and then to (4.8). We obtain (4.9) in a similar way. Thus, (4.4)

holds for anym, n≥0.

Acknowledgements. The author thanks Dr. Chris Tisdell for his constructive and thorough feedback on this manuscript, and Dr. Douglas Anderson for his useful comments.

References

[1] R. Agarwal, M. Bohner, D. O’Regan and A. Peterson;Dynamic equations on time scales: a survey, J. Comput. Appl. Math.141(2002), No. 1–2, 1–26.

[2] R. P. Agarwal and V. Lakshmikantham;Uniqueness and nonuniqueness criteria for ordinary differential equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1993.

[3] R. P. Agarwal, D. O’Regan, V. Lakshmikantham and S. Leela;Existence of positive solutions for singular initial and boundary value problems via the classical upper and lower solution approach, Nonlinear Anal.50(2002), No. 2, Ser. A: Theory Methods, 215–222.

[4] E. Akin;Boundary value problems for a differential equation on a measure chain, Panamer.

Math. J.10(2000), No. 3, 17–30.

[5] E. Akın-Bohner, F. M. Atıcı, and B. Kaymak¸calan;Lower and upper solutions of boundary value problems, In: “Advances in Dynamic Equations on Time Scales”, M. Bohner and A.

Peterson, editors, 165–188. Birkh¨auser, Boston, 2003.

(13)

[6] M. F. Atici and D. C. Biles;First- and second-order dynamic equations with impulse, Adv.

Difference Equ.2(2005) 119–132.

[7] M. F. Atici and D. C. Biles;First order dynamic inclusions on time scales, J. Math. Anal.

Appl.292(2004), No. 1, 222–237.

[8] M. F. Atici and G. Sh. Guseinov;On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math.141(2002), No. 1–2, 75–99.

[9] M. Bohner and A. Peterson;Dynamic equations on time scales. An introduction with applications, Birkh¨auser Boston, Inc., Boston, MA, 2001.

[10] M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Birkh¨auser Boston, Inc., Boston, MA, 2003.

[11] E. A. Coddington;An introduction to ordinary differential equations, Prentice-Hall Mathe- matics Series, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1961.

[12] S. Hilger;Analysis on measure chains - a unified approach to continuous and discrete calculus, Res. Math.18(1990), 18–56.

[13] S. Hilger;Differential and difference calculus–unified!, Proceedings of the Second World Con- gress of Nonlinear Analysts, Part 5 (Athens, 1996), Nonlinear Anal. Series A: Theory and Methods,30(1997), No. 5, 2683–2694.

[14] B. Kaymak¸calan and B. A. Lawrence;Coupled solutions and monotone iterative techniques for some nonlinear initial value problems on time scales, Nonlinear Anal. Real World Appl.

4(2003), No. 2, 245–259.

[15] V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan;Dynamic systems on measure chains, Mathematics and its Applications, No. 370, Kluwer Academic Publishers Group, 1996.

[16] J. LaSalle; Uniqueness theorems and successive approximations, Ann. of Math.50(1949), No. 2, 722–730.

[17] N. G. Lloyd;Degree theory, Cambridge Tracts in Mathematics, No. 73, Cambridge University Press, Cambridge, 1978.

[18] Donal O’Regan;Existence theory for nonlinear ordinary differential equations, Kluwer Aca- demic Publishers Group, Dordrecht, 1997.

[19] C. C. Tisdell and A. H. Zaidi.Successive approximation to solutions of dynamic equations on time scales, Comm. Appl. Nonlinear Anal.16(2009), No. 1, 61–87.

[20] C. C. Tisdell and A. Zaidi.Basic qualitative and quantitative results for solutions to nonlinear dynamic equations on time scales with an application to economic modelling, Nonlinear Anal.

68(2008), No 11, 3504–3524.

[21] B. Q. Yan, D. O’Regan and Ravi P. Agarwal;Positive solutions to singular boundary value problems with sign changing nonlinearities on the half-line via upper and lower solutions, Acta Math. Sin. (Engl. Ser.)23(2007), No. 8, 1447–1456.

[22] E. Zeidler;Nonlinear functional analysis and its applications. Fixed-point theorems, Trans- lated from the German by Peter R. Wadsack. Springer-Verlag, New York, 1986.

Atiya H. Zaidi

School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia

E-mail address:[email protected]